Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras

The conformal Galilei algebra (CGA) is a non-semisimple Lie algebra labelled by two parameters d and ℓ. The aim of the present work is to investigate the lowest weight representations of CGA with d=1 for any integer value of ℓ. First we focus on the reducibility of the Verma modules. We give a formu...

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Дата:2015
Автори: Aizawa, N., Chandrashekar, R., Segar, J.
Формат: Стаття
Мова:English
Опубліковано: Інститут математики НАН України 2015
Назва видання:Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:http://dspace.nbuv.gov.ua/handle/123456789/146864
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Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras / N. Aizawa, R. Chandrashekar, J. Segar // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 39 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
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spelling irk-123456789-1468642019-02-12T01:23:52Z Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras Aizawa, N. Chandrashekar, R. Segar, J. The conformal Galilei algebra (CGA) is a non-semisimple Lie algebra labelled by two parameters d and ℓ. The aim of the present work is to investigate the lowest weight representations of CGA with d=1 for any integer value of ℓ. First we focus on the reducibility of the Verma modules. We give a formula for the Shapovalov determinant and it follows that the Verma module is irreducible if ℓ=1 and the lowest weight is nonvanishing. We prove that the Verma modules contain many singular vectors, i.e., they are reducible when ℓ≠1. Using the singular vectors, hierarchies of partial differential equations defined on the group manifold are derived. The differential equations are invariant under the kinematical transformation generated by CGA. Finally we construct irreducible lowest weight modules obtained from the reducible Verma modules. 2015 Article Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras / N. Aizawa, R. Chandrashekar, J. Segar // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 39 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B10; 58J70 DOI:10.3842/SIGMA.2015.002 http://dspace.nbuv.gov.ua/handle/123456789/146864 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description The conformal Galilei algebra (CGA) is a non-semisimple Lie algebra labelled by two parameters d and ℓ. The aim of the present work is to investigate the lowest weight representations of CGA with d=1 for any integer value of ℓ. First we focus on the reducibility of the Verma modules. We give a formula for the Shapovalov determinant and it follows that the Verma module is irreducible if ℓ=1 and the lowest weight is nonvanishing. We prove that the Verma modules contain many singular vectors, i.e., they are reducible when ℓ≠1. Using the singular vectors, hierarchies of partial differential equations defined on the group manifold are derived. The differential equations are invariant under the kinematical transformation generated by CGA. Finally we construct irreducible lowest weight modules obtained from the reducible Verma modules.
format Article
author Aizawa, N.
Chandrashekar, R.
Segar, J.
spellingShingle Aizawa, N.
Chandrashekar, R.
Segar, J.
Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras
Symmetry, Integrability and Geometry: Methods and Applications
author_facet Aizawa, N.
Chandrashekar, R.
Segar, J.
author_sort Aizawa, N.
title Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras
title_short Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras
title_full Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras
title_fullStr Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras
title_full_unstemmed Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras
title_sort lowest weight representations, singular vectors and invariant equations for a class of conformal galilei algebras
publisher Інститут математики НАН України
publishDate 2015
url http://dspace.nbuv.gov.ua/handle/123456789/146864
citation_txt Lowest Weight Representations, Singular Vectors and Invariant Equations for a Class of Conformal Galilei Algebras / N. Aizawa, R. Chandrashekar, J. Segar // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 39 назв. — англ.
series Symmetry, Integrability and Geometry: Methods and Applications
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AT chandrashekarr lowestweightrepresentationssingularvectorsandinvariantequationsforaclassofconformalgalileialgebras
AT segarj lowestweightrepresentationssingularvectorsandinvariantequationsforaclassofconformalgalileialgebras
first_indexed 2023-05-20T17:25:54Z
last_indexed 2023-05-20T17:25:54Z
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